## A Dipole in front of a flat Metallic surface

The Green's function can be used to find the electric field or electrostatic potential for arbitrary charge distributions. You will learn that the polarization of a dipole in front of a flat metallic surface enormously affects the overall field.

## Problem Statement

Use the Green's function to calculate the electrostatic potential $$\phi\left(\mathbf{r}\right)$$  for a dipole $$\mathbf{p}$$ at $$\mathbf{r}_{D}$$ in front of a flat metallic surface terminated at $$z=0$$. Determine  the main contributions to the potential for $$\left|\mathbf{r}_{D}\right|\ll\left|\mathbf{r}\right|$$. Which has a stronger fall-of: $$x$$- or $$z$$-polarization? Can you explain this in simple words?

## Background: Dipole Emitters in Front of Flat Metallic Surfaces

In this problem, we discuss the electrostatic dipole in the vicinity of a flat metal which is of huge interest both in applied and fundamental physics: think of a dipole antenna with a metal backplate or an emitting molecule. Of course such systems demand an understanding of full electrodynamics. The point is, however, that the main features will remain; $$x$$- and $$z$$-polarization will radiate differently.

It gets even more interesting if we do not have a perfect metal. Then, the electric field can penetrate the conductor and a closely placed dipole can excite surface waves as predicted by Sommerfeld more than one hundred years ago. Even though this effect was not the explanation for long-range radio transmissions, the theory of Sommerfeld is still at the core of plasmonics, the field that deals with these so-called surface plasmon polaritons. For much more information on dipoles in front of flat metallic surfaces please have a look at Lukas Novotny's freely available book chapter on the topic.